# Pendulum patterns

May 13, 2011 at 10:47 PM by Dr. Drang

It was on Boing Boing and Kottke, so I’m sure you’ve seen this video of 15 tuned pendulums.

The pendulum lengths are set so longest pendulum swings through 51 cycles per minute, the second-longest 52 cycles per minute, and so on up through 65 cycles per minute. As we showed in this post a couple of years ago, the relationship between period and length is

$$T=2\pi \sqrt{\frac{L}{g}}$$or

$$L=\frac{g\phantom{\rule{0.278em}{0ex}}{T}^{2}}{4{\pi}^{2}}$$So to get a pendulum with a period of $\frac{60}{51}$ seconds, the length should be

$$L=\frac{9807(\frac{60}{51}{)}^{2}}{4{\pi}^{2}}=344\phantom{\rule{0.222em}{0ex}}\mathrm{m}\mathrm{m}$$which looks about right for what we see in the video. Here’s a table with all the lengths needed to make the device:

Cycles/min | Length (mm) |
---|---|

51 | 344 |

52 | 331 |

53 | 318 |

54 | 307 |

55 | 296 |

56 | 285 |

57 | 275 |

58 | 266 |

59 | 257 |

60 | 248 |

61 | 240 |

62 | 233 |

63 | 225 |

64 | 218 |

65 | 212 |

Note that the pendulum length is not the string length, it’s the vertical distance from the top bar to the center of the bob.

I wondered what it would look like to graph a minute’s worth of oscillation for each of the 15 pendulums on the same plot. How would the patterns we see in the video appear in the graph? So I wrote this little gnuplot script to find out.

```
1: set terminal postscript solid
2: set output "pendulum.ps"
3: set xrange[0:60]
4: set yrange[-1:1]
5: unset key
6: unset border
7: unset xtics
8: unset ytics
9: set size ratio .1
10:
11: set samples 2400
12:
13: plot cos(2*pi*51/60*x) w l lc -1,\
14: cos(2*pi*52/60*x) w l lc -1,\
15: cos(2*pi*53/60*x) w l lc -1,\
16: cos(2*pi*54/60*x) w l lc -1,\
17: cos(2*pi*55/60*x) w l lc -1,\
18: cos(2*pi*56/60*x) w l lc -1,\
19: cos(2*pi*57/60*x) w l lc -1,\
20: cos(2*pi*58/60*x) w l lc -1,\
21: cos(2*pi*59/60*x) w l lc -1,\
22: cos(2*pi*60/60*x) w l lc -1,\
23: cos(2*pi*61/60*x) w l lc -1,\
24: cos(2*pi*62/60*x) w l lc -1,\
25: cos(2*pi*63/60*x) w l lc -1,\
26: cos(2*pi*64/60*x) w l lc -1,\
27: cos(2*pi*65/60*x) w l lc -1
```

I used Preview to convert the PostScript output to a PNG file and rotated it to a vertical orientation so time increases as we move down the plot (you can click on the image to see a bigger version).

Not surprisingly, the graph is just as patterned as the video. If I were more ambitious, I’d annotate the graph with timestamps from the video to point out interesting portions.